Is Calculus a Smooth Transition After Mastering Trig and Algebra 2?

[Brady]

Hi, I'm in the complex numbers section of a trig book, and I'm having trouble intuitively understanding how a number like 3+5i can become (3,5) on the Gaussian coordinate plane...the logic behind it doesn't jump out at me...

Any help?

And is calculus generally a smooth transition after mastering trig and algebra 2, or is it something totally different?

Thanks a lot,
Brady Yoon

 

[quasar987]

Hi, I'm in the complex numbers section of a trig book, and I'm having trouble intuitively understanding how a number like 3+5i can become (3,5) on the Gaussian coordinate plane...the logic behind it doesn't jump out at me...

ummm. The idea of the Gaussian plane is to represent complex numbers as vectors. We chose the x-axis to host the real part of the complex number and the y-axis to host the imaginary part. Thus a complex number a+bi has the vector representation (a,b) in the Gaussian plane.


though you might not completely understand this, I will add that $\mathbb{C}$ and $\mathbb{R}^2$, as groups, are isomorphic to one another [according to the isomorphism that assigns to each complex number its corresponding vector in the Gaussian plane: f(a+bi) = (a,b)].
Maybe someone else can extrapolate on what intesresting things this implies; I'd be interested. Thx.

 

[mathmike]

a smooth translation from trig is non existent. calculus is just a lot of algebra with a few more formulas thrown in. don't let it scare you though, it isn't as hard as you think

 

[HallsofIvy]

That's just a way of representing the complex numbers. Just as we can think of real numbers as numbers on a "number line", since every complex number, a+ bi, requires two real numbers, we need two number lines to represent complex number. It happens to be simplest to make those number lines perpendicular. The complex number a+bi is represented by the pair (a,b) in an obvious way and that corresponds to the point with coordinates (a, b).